Quantum Incompatibility Witnesses

We demonstrate that quantum incompatibility can always be detected by means of a state discrimination task with partial intermediate information. This is done by showing that only incompatible measurements allow for an efficient use of premeasurement information in order to improve the probability of guessing the correct state. Thus, the gap between the guessing probabilities with pre- and postmeasurement information is a witness of the incompatibility of a given collection of measurements. We prove that all linear incompatibility witnesses can be implemented as some state discrimination protocol according to this scheme. As an application, we characterize the joint measurability region of two noisy mutually unbiased bases.

Figure 1

Witnesses are associated with hyperplanes, and they are detection equivalent if they yield the same separation of the set $C$. Here, two tight equivalent witnesses detect the red point, but not the black one.

Figure 2

The set of $\mathit{\gamma }=({\gamma }_{\phi },{\gamma }_{\psi })$ for which Eq. (10) defines two measurements (green square), and the one for which these measurements are compatible (blue region) for different values of the dimension $d$. The red line is the curve (12). The case $d=2$ is special, and was already treated in [35].